No abstract

An important problem in spatial ecology is to understand how population-scale patterns emerge from individual-level birth, death, and movement processes. These processes, which depend on local landscape characteristics, vary spatially and may exhibit sharp transitions through behavioural responses to habitat edges, leading to discontinuous population densities. Such systems can be modelled using reaction-diffusion equations with interface conditions that capture local behaviour at patch boundaries. In this work we develop a novel homogenization technique to approximate the large-scale dynamics of the system. We illustrate our approach, which also generalizes to multiple species, with an example of logistic growth within a periodic environment. We find that population persistence and the large-scale population carrying capacity is influenced by patch residence times that depend on patch preference, as well as movement rates in adjacent patches. The forms of the homogenized coefficients yield key theoretical insights into how large-scale dynamics arise from the small-scale features.

Phenology, the timing of developmental events such as oviposition or pupation, is highly dependent on temperature; since insects are ectotherms, the time it takes them to complete a life stage (development time) depends on the temperatures they experience. This dependence varies within and between populations due to variation among individuals that is fixed within a life stage (giving rise to what we call persistent variation) and variation from random effects within a life stage (giving rise to what we call random variation). It is important to understand how both types of variation affect phenology if we are to predict the effects of climate change on insect populations.We present three nested phenology models incorporating increasing levels of variation. First, we derive an advection equation to describe the temperature-dependent development of a population with no variation in development time. This model is extended to incorporate persistent variation by introducing a developmental phenotype that varies within a population, yielding a phenotype-dependent advection equation. This is further extended by including a diffusion term describing random variation in a phenotype-dependent Fokker-Planck development equation. These models are also novel because they are formulated in terms of development time rather than developmental rate; development time can be measured directly in the laboratory, whereas developmental rate is calculated by transforming laboratory data. We fit the phenology models to development time data for mountain pine beetles (MPB) (Dendroctonus ponderosae Hopkins [Coleoptera: Scolytidae]) held at constant temperatures in laboratory experiments. The nested models are parameterized using a maximum likelihood approach. The results of the parameterization show that the phenotype-dependent advection model provides the best fit to laboratory data, suggesting that MPB phenology may be adequately described in terms of persistent variation alone. MPB phenology is simulated using phloem temperatures and attack time distributions measured in central Idaho. The resulting emergence time distributions compare favorably to field observations.

Climate change is likely to disrupt the timing of developmental events (phenology) in insect populations in which development time is largely determined by temperature. Shifting phenology puts insects at risk of being exposed to seasonal weather extremes during sensitive life stages and losing synchrony with biotic resources. Additionally, warming may result in loss of developmental synchronization within a population making it difficult to find mates or mount mass attacks against well-defended resources at low population densities. It is unknown whether genetic evolution of development time can occur rapidly enough to moderate these effects. We present a novel approach to modeling the evolution of phenology by allowing the parameters of a phenology model to evolve in response to selection on emergence time and density. We use the Laplace method to find asymptotic approximations for the temporal variation in mean phenotype and phenotypic variance arising in the evolution model that are used to characterize invariant distributions of the model under periodic temperatures at leading order. At these steady distributions the mean phenotype allows for parents and offspring to be oviposited at the same time of year in consecutive years. Numerical simulations show that populations evolve to these steady distributions under periodic temperatures. We consider an example of how the evolution model predicts populations will evolve in response to warming temperatures and shifting resource phenology.

Ecologists are often challenged by complex scale relationships: we want to understand and predict population dynamics on large spatial and temporal scales, but we can typically only measure and observe processes on small scales. Methods exist to obtain local population growth and movement models from observational data (e.g. Turchin, 1998). Theory exists to calculate, say, spread rates and persistence conditions from homogeneous landscape-level models (e.g. Fagan et al., 2002). How to obtain landscape-level models from habitat-level information is the gap that we bridge here (Figure 1). Our approach is based on homogenization, a mathematical theory of coarse-graining or weighted averaging, with weights that emerge from the theory. It applies when individuals, over their lifetimes, experience several different habitat types that affect their movement behaviours and vital rates. Our formulas give precise relationships for how to include small-scale variation into landscape-scale models. Our method can be used to develop new understanding of mechanisms that drive large-scale population outcomes, and also to connect increasingly available